GrassmannCalculus | Wolfram Resource Object

GrassmannCalculus`

ComposePlaneElement

ComposePlaneElement[c]

composes a plane element as the product of a point and two vectors. ComposePlaneElement[c] has an alias Subscript[★Π,c].

ComposePlaneElement[c,s]

starts the subscript indicies at

s

.

Details and Options



Examples

(1)

Basic Examples

(1)

In[1]:=

<<GrassmannCalculus`

Set the book 3-dimensional Point space.

In[2]:=

★A;

★

3

;

Compare a point, line and plane element in 3-space.

In[3]:=

MapThreadRow[{#1,#2[c]}]&,{"Point: ","Line: ","Plane: "},

ComposePoint

,

ComposeLineElement

,

ComposePlaneElement

//Column

Out[3]=

Point: ★+

c

1

e

1

+

c

2

e

2

+

c

3

e

3

Line: (★+

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)

Plane: (★+

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)⋀(

c

7

e

1

+

c

8

e

2

+

c

9

e

3

)

You can use the full form of

ComposePlaneElement[c]

, or its alias

★Π

c

.

In[4]:=

★A;

★

3

;

ComposePlaneElement

[c]

,

★Π

c

//Column

Out[4]=

(★+

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)⋀(

c

7

e

1

+

c

8

e

2

+

c

9

e

3

)

(★+

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)⋀(

c

7

e

1

+

c

8

e

2

+

c

9

e

3

)

The current space needs to be a point space. If it is not, no composition will occur.

In[5]:=



★ℬ

3

;

★Π

a

,

★

3

;

★Π

a

//Column

Out[5]=

ComposePlaneElement[a]

(★+

a

1

e

1

+

a

2

e

2

+

a

3

e

3

)⋀(

a

4

e

1

+

a

5

e

2

+

a

6

e

3

)⋀(

a

7

e

1

+

a

8

e

2

+

a

9

e

3

)

You can compose plane elements in any dimension.

In[6]:=

Table

★

i

;

★Π

c

,{i,1,4}//Column

Out[6]=

(★+c

e

1

)⋀(

c

2

e

1

)⋀(

c

3

e

1

)

(★+

c

1

e

1

+

c

2

e

2

)⋀(

c

3

e

1

+

c

4

e

2

)⋀(

c

5

e

1

+

c

6

e

2

)

(★+

c

1

e

1

+

c

2

e

2

+

c

3

e

3

)⋀(

c

4

e

1

+

c

5

e

2

+

c

6

e

3

)⋀(

c

7

e

1

+

c

8

e

2

+

c

9

e

3

)

(★+

c

1

e

1

+

c

2

e

2

+

c

3

e

3

+

c

4

e

4

)⋀(

c

5

e

1

+

c

6

e

2

+

c

7

e

3

+

c

8

e

4

)⋀(

c

9

e

1

+

c

10

e

2

+

c

11

e

3

+

c

12

e

4

)

You can compose lists of plane elements.

In[7]:=

★

2

;

★Π

{a,b,c}

//Column

Out[7]=

(★+

a

1

e

1

+

a

2

e

2

)⋀(

a

3

e

1

+

a

4

e

2

)⋀(

a

5

e

1

+

a

6

e

2

)

(★+

b

1

e

1

+

b

2

e

2

)⋀(

b

3

e

1

+

b

4

e

2

)⋀(

b

5

e

1

+

b

6

e

2

)

(★+

c

1

e

1

+

c

2

e

2

)⋀(

c

3

e

1

+

c

4

e

2

)⋀(

c

5

e

1

+

c

6

e

2

)

You can make the starting index of the coefficients whatever you want.

In[8]:=

★Π

{a,b,c},0

//Column

Out[8]=

(★+

a

0

e

1

+

a

1

e

2

)⋀(

a

2

e

1

+

a

3

e

2

)⋀(

a

4

e

1

+

a

5

e

2

)

(★+

b

0

e

1

+

b

1

e

2

)⋀(

b

2

e

1

+

b

3

e

2

)⋀(

b

4

e

1

+

b

5

e

2

)

(★+

c

0

e

1

+

c

1

e

2

)⋀(

c

2

e

1

+

c

3

e

2

)⋀(

c

4

e

1

+

c

5

e

2

)

In[9]:=

★Π

{a,b,c},{-2,0,α}

//Column

Out[9]=

(★+

a

-2

e

1

+

a

-1

e

2

)⋀(

a

0

e

1

+

a

1

e

2

)⋀(

a

2

e

1

+

a

3

e

2

)

(★+

b

0

e

1

+

b

1

e

2

)⋀(

b

2

e

1

+

b

3

e

2

)⋀(

b

4

e

1

+

b

5

e

2

)

(★+

c

α

e

1

+

c

1+α

e

2

)⋀(

c

2+α

e

1

+

c

3+α

e

2

)⋀(

c

4+α

e

1

+

c

5+α

e

2

)

Note that all the scalar symbols generated as coefficients of the basis elements have been automatically declared as scalar symbols.

In[10]:=

ScalarSymbols

Out[10]=

{a,b,c,d,e,f,g,h,

a

-2

,

a

-1

,

a

0

,

a

1

,

a

2

,

a

3

,

a

4

,

a

5

,

a

6

,

b

0

,

b

1

,

b

2

,

b

3

,

b

4

,

b

5

,

b

6

,

c

0

,

c

1

,

c

2

,

c

3

,

c

4

,

c

5

,

c

6

,

c

α

,

c

1+α

,

c

2+α

,

c

3+α

,

c

4+α

,

c

5+α

}

You can also generate templates for plane elements using the placeholder symbol. This allows you to tab through the composed result and enter your own values.

In[11]:=

★

3

;

★Π



Out[11]=

(★+

e

1

+

e

2

+

e

3

)⋀(

e

1

+

e

2

+

e

3

)⋀(

e

1

+

e

2

+

e

3

)

This could be copied and filled in to create a specific plane.

In[12]:=

(★+1

e

1

+2

e

2

+3

e

3

)⋀(4

e

1

+5

e

2

+6

e

3

)⋀(a

e

1

+b

e

2

+c

e

3

)

Out[12]=

(★+

e

1

+2

e

2

+3

e

3

)⋀(4

e

1

+5

e

2

+6

e

3

)⋀(a

e

1

+b

e

2

+c

e

3

)

Subscripted scalars are convenient for generic derivations. For specific cases it will often be easier simply to dot the Basis with a List of coefficients.

In[13]:=

({1,1,2,3}.Basis)⋀({0,4,5,6}.Basis)⋀({0,a,b,c}.Basis)

Out[13]=

(★+

e

1

+2

e

2

+3

e

3

)⋀(4

e

1

+5

e

2

+6

e

3

)⋀(a

e

1

+b

e

2

+c

e

3

)

It is also possible to enter a plane element as, say, two points and a vector using the point and vector composers within an exterior product.

In[14]:=

★



⋀

★



⋀

★



Out[14]=

(★+

e

1

+

e

2

+

e

3

)⋀(★+

e

1

+

e

2

+

e

3

)⋀(

e

1

+

e

2

+

e

3

)

SeeAlso

▪

▪

▪

▪

RelatedGuides

▪

Composers

""